Prospective K-8 Teachers’ Knowledge of Relational Thinking

The goal of this study was to examine two issues: First, pre-service teachers’ ability and inclination to think relationally prior to instruction about the role relational thinking plays in the K-8 mathematics curriculum. Second, to examine task specific variables possibly associated with pre-service teachers’ inclination to engage in relational thinking. The results revealed that preservice teachers engage in relational thinking about equality, however, their inclination to do so is rather limited. Furthermore, they tend to engage in relational thinking more frequently in the context of arithmetic than algebra-related tasks. Pre-service teachers’ inclination to engage in relational thinking appeared to also relate to the overall task complexity and the use of variables. Implications of these findings for pre-service teacher education are provided

Engaging with issues of emotionality in mathematics teacher education for social justice

This article focuses on the relationship between social justice, emotionality and mathematics teaching in the context of the education of prospective teachers of mathematics. A relational approach to social justice calls for giving attention to enacting socially-just relationships in mathematics classrooms. Emotionality and social justice in teaching mathematics variously intersect, interrelate or interweave. An intervention, usng creative action methods, with a cohort of prospective teachers addressing these issues is described to illustrate the connection between emotionality and social justice in the context of mathematics teacher education. Creative action methods involve a variety of dramatic, interactive and experiential tools that can promote personal and group engagement and embodied reflection. The intervention aimed to engage the prospective teachers with some key issues for social justice in mathematics education through dialogue about the emotionality of teaching and learning mathematics. Some of the possibilities and limits of using such methods are considered

An extension of relational methods in mortality estimations

Actuaries and demographers have a long tradition of utilising collateral data to improve mortality estimates. Three main approaches have been used to accomplish the improvement- mortality laws, model life tables, and relational methods. The present paper introduces a regression model that incorporates all of the beneficial principles from each of these approaches. The model is demonstrated on mortality data pertaining to various groups of life insured people in Sweden.applied mathematics, collateral data, model life table, mortality law, relational method

Developing Learning Trajectory For Enhancing Students’ Relational Thinking

Algebra is part of Mathematics learning in Indonesian curriculum. It takes one half of the teaching hours in senior high school, and one third in junior high school. However, the learning trajectory of Algebra needs to be improved because teachers teach computational thinking by applying paper-and-pencil strategy combining with the concepts-operations-example-drilling approach. Mathematics textbooks do not give enough guidance for teachers to conduct good activities in the classroom to promote algebraic thinking especially in the primary schools. To reach Indonesian Mathematics teaching goals, teachers should develop learning trajectories based on pedagogical and theoretical backgrounds. Teachers need to have knowledge of student’s developmental progressions and understanding of mathematics concepts and students’ thinking. Research shows that teachers’ knowledge of student’s mathematical development is related to their students’ achievement. In fostering a greater emphasis on algebraic thinking, teachers and textbooks need to work more closely together to develop a clearer learning trajectory. Having this algebraic thinking, students developed not only their own character of learning and thinking but also their attitude, attention and discipline. Key Words: Learning Trajectory, Relational Thinkin

Building Conceptual Understandings of Equivalence

The equal sign is prevalent at all levels of mathematics however many students misunderstand the meaning of the equal sign and consider it an operational symbol for the completion of an algorithm (Baroody & Ginsburg, 1983; Rittle-Johnson & Alibali, 1999). Three constructs were studied through the lens of the Developing Mathematical Thinking (Brendefur, 2008), Relational Thinking, Spatial Reasoning and Modes of Representation. A review of literature was conducted to examine the effects of mathematics instruction on the development of students’ conceptual understanding of equivalence through the integration of spatial reasoning and relational thinking. The Developing Mathematical Thinking (DMT) curricular resources integrate Bruner’s enactive, iconic, and symbolic modes of representations (1966), using tasks designed to strengthen students’ spatial reasoning and relational thinking to develop mathematical equivalence. The research question “What is the effect of integrating iconic teaching methodology into mathematics instruction on first grade students’ relational thinking and spatial reasoning performance?” was analyzed to determine whether there was a significant difference in pre-and posttest scores for the two groups. Students were found to have a better opportunity to develop conceptual understanding of mathematics in their early years of school when taught with the progression of EIS, relational thinking and spatial reasoning

Polystore mathematics of relational algebra

Financial transactions, internet search, and data analysis are all placing increasing demands on databases. SQL, NoSQL, and NewSQL databases have been developed to meet these demands and each offers unique benefits. SQL, NoSQL, and NewSQL databases also rely on different underlying mathematical models. Polystores seek to provide a mechanism to allow applications to transparently achieve the benefits of diverse databases while insulating applications from the details of these databases. Integrating the underlying mathematics of these diverse databases can be an important enabler for polystores as it enables effective reasoning across different databases. Associative arrays provide a common approach for the mathematics of polystores by encompassing the mathematics found in different databases: sets (SQL), graphs (NoSQL), and matrices (NewSQL). Prior work presented the SQL relational model in terms of associative arrays and identified key mathematical properties that are preserved within SQL. This work provides the rigorous mathematical definitions, lemmas, and theorems underlying these properties. Specifically, SQL Relational Algebra deals primarily with relations - multisets of tuples - and operations on and between those relations. These relations can be modeled as associative arrays by treating tuples as non-zero rows in an array. Operations in relational algebra are built as compositions of standard operations on associative arrays which mirror their matrix counterparts. These constructions provide insight into how relational algebra can be handled via array operations. As an example application, the composition of two projection operations is shown to also be a projection, and the projection of a union is shown to be equal to the union of the projections